Analytic compactifications of C^2 part II - one irreducible curve at infinity

TL;DR

This paper classifies primitive normal compactifications of C^2 with an irreducible curve at infinity, analyzes their moduli and automorphisms, and applies these results to classify certain algebraic surfaces and weighted projective spaces.

Contribution

It provides a comprehensive classification of primitive normal compactifications of C^2 with an irreducible curve at infinity and explores their moduli, automorphisms, and applications to algebraic surface classification.

Findings

Most such surfaces have C^2 rigidly embedded.

Computed the canonical divisor and related log discrepancy to Frobenius number.

Characterized weighted projective spaces P^2(1,1,q) via log discrepancy and index.

Abstract

We classify 'primitive normal compactifications' of C^2 (i.e. normal analytic surfaces containing C^2 for which the curve at infinity is irreducible), compute the moduli space of these surfaces and their groups of auomorphisms. In particular we show that in 'most' of these surfaces C^2 is 'rigidly embedded'. As an application we give a description of 'embedded isomorphism classes' of planar curves with one place at infinity. We also compute the canonical divisor of these surfaces; it turns out that their log discrepancy is related to the Frobenius number of the semigroup of poles along the curve at infinity. We use the computation to classify Gorenstein primitive compactifications of C^2 with rational and minimally elliptic singularities, extending a result of Brenton, Drucker and Prins (Ann. of Math. Stud., vol 100, 1981). As another application we characterize weighted projective spaces of the form P^2(1,1,q) in terms of their 'log discrepancy' and 'index', generalizing a characterization of P^2 due to Borisov (Journal of Algebraic Combinatorics, 2014).